Abstract
It’s universally acknowledged that the Helium 4 superfluid has soliton solutions. In recent researches, we can also observe the dynamic evolutions of ultracold atoms in an optical superlattice system. It seems to be predictable that there are also soliton solutions in the condition of soft-core bosons, but if the interaction among bosons is quite huge, it will become tricky to ascertain if hard-core bosons also have solutions of soliton. Our topic is to find out if there are soliton solutions that stably transport messages when hard-core bosons couple with other systems in an optical superlattice. It’s critical to engineering and science, because all the real-life systems are the couplings of hard-core bosons and other systems, and there are also demonstrations which figure out that hard-core models are more accurate than soft-core models. The first chapter of this paper will make a brief introduction to the basic models of soliton and their origins in a variety of mathematical models. In the second chapter we will see the soliton solutions of soft-core bosons in superfluid and prove that soliton solutions do not exist under critical speed. In chapter 3, a review toward the research results in the past 3 years will be performed, and we will use spin-coherent state representation to solve the problem-the soliton solutionsof hard-core bosons in optical superlattice. The result we get is not intuitive– there are bright solitons that can transport messages stably under critical speed. In chapter 4, we will add another kind of fermions, let their spin state couple with bose gases, investigate what result we will get when we use this novel way to couple two gases, and then point out the limitation of this way and problems waiting for solutions in the future. We will also show that as long as discrete Hamiltonian has direct exchange interaction term, it will be able to have soliton solutions near the phase transition point of condensed matter. This is the most critical point of transportation problem– solutions that can stably transport messages do exist.